Single Variable Calculus, Chapter 3, 3.7, Section 3.7, Problem 17

Suppose that the mass of the part of a metal rod that lies between its left end and a point $x$ meters to the right is $3x^2$ kg. Find the linear density when $x$ is (a) 1m, (b) 2m, and (c) 3m. Where is the density highest? Slowest?


$
\begin{equation}
\begin{aligned}
\text{Linear Density } = \frac{dm}{dx} &= 3 \frac{d}{dx}(x^2)\\
\\
\frac{dm}{dx} &= 3(2x)\\
\\
\frac{dm}{dx} &= 6x
\end{aligned}
\end{equation}
$


a.) when $x = 1m$,
$\displaystyle \frac{dm}{dx} = 6(1) = 6 \frac{\text{kg}}{\text{m}}$

b.) when $x = 2m$,
$\displaystyle \frac{dm}{dx} = 6(2) = 12 \frac{\text{kg}}{\text{m}}$

c.) when $x = 3m$,
$\displaystyle \frac{dm}{dx} = 6(3) = 18 \frac{\text{kg}}{\text{m}}$

Based from the values we obtain, the density is highest when $x = 3m$. On the other hand, the density is lowest at $x =1m$. It means that the density is highest at the right end of the rod while the density is lowest at the left end of the rod.